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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/The Tangent and Velocity Problems">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Mon Feb 10 16:39:31 1997}
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\section{The Tangent\label{Tangent} and Velocity Problems}

The word \textbf{tangent} is derived from the Latin word \emph{tangens},
which means ``touching.'' Thus a tangent to a curve is a line that touches
the curve. The tangent to a curve at a point $\left( x,y\right) $ on the
curve is, in some real sense, the best possible approximation to the curve
at that point by a straight line. In Euclidean geometry, the tangent to a
circle is defined as a line that has exactly one point in common with the
circle. For general curves this definition is not adequate---the tangent
line to a straight line has infinitely many points in common with the line,
for example! The tangent line (if it exists) is found as the limit of a
sequence of approximations by secants, which are lines passing through two
points of the curve. The sequence is obtained by fixing the point in
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\subsubsection{The Velocity Problem\label{Velocity problem}}

How is ``instantaneous velocity'' defined? If you measure the time taken to
move from one point to another you can calculus the average velocity by
dividing the distance traversed by the time. You can approximate the
instantaneous velocity by fixing the starting point measuring the time taken
to travel to successively closer points.

This is analogous to the tangent problem posed above, where you could find
the slope of secant lines---lines that passed through two points on the
curve---and approximate the slope of a tangent that passed through just one
point.

\subsubsection{Zooming In for a Closer Look at the Problem\label{Zoom tools}}

To shed light on the idea of approximating a curve by a straight line, try
plotting a curve and zooming in with a computer graphing system or your
graphing calculator for a closer and closer look at the curve around the
point in question. You will quickly see that when you get close enough to
the point the curve appears as a straight line. This gives you a visual
glimpse of the slope of the tangent line at that point.

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Zoom plotting tools}{}{}{../../Help/DM6-3.tex#Zoom}}

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